seurat_dependency (SAVER)

1. Import Data and Library

library(Seurat)

## Attaching SeuratObject

library(SeuratData)

## Registered S3 method overwritten by 'cli':
##   method     from         
##   print.boxx spatstat.geom

## ── Installed datasets ───────────────────────────────────── SeuratData v0.2.1 ──

## ✓ bmcite       0.3.0                    ✓ pbmcMultiome 0.1.0

## ────────────────────────────────────── Key ─────────────────────────────────────

## ✓ Dataset loaded successfully
## > Dataset built with a newer version of Seurat than installed
## ❓ Unknown version of Seurat installed

library(cowplot)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(SAVER)

# Just an example way to preprocess data, to demonstrate the PCA part
# # You could use SCTransform into RunPCA also
# bm <- LoadData(ds = "bmcite")
load("SAVER_seurat.Rdata")

orig_count <- Matrix::t(bm@assays$RNA@counts)
celltype_factor <- as.numeric(as.factor(bm@meta.data$celltype.l2))
# save(bm, saver_dat, cell_type, sub_dat, sub_celltype, store_date, file = "SAVER_seurat.Rdata")

cell_labels <- unique(bm@meta.data$celltype.l2)
# cell_type <- bm@meta.data$celltype.l2
# rand_ind <- c()
# 
# for (cell in cell_labels){
#   set.seed(10)
#   subcell_ind <- which(cell_type == cell)
#   subcell_len <- length(subcell_ind)
#   subcell_mat <- orig_count[subcell_ind, ]
# 
#   row_ind <- apply(orig_count, 1, function(x){length(which(x != 0))})
#   idx <- order(row_ind, decreasing = T)
#   sub_rand <- sample(length(subcell_ind),
#                      length(subcell_ind)/40)
#   rand_ind <- c(rand_ind, idx[1:(subcell_len/30)])
# }
# 
# sub_dat <- orig_count[rand_ind, ]
# 
# col_ind <- apply(sub_dat, 2, function(x){length(which(x != 0))})
# idx <- order(col_ind, decreasing = T)[1:500]
# 
# sub_dat <- sub_dat[, idx]
# 
# dat_hclust <- hclust(dist(t(as.matrix(sub_dat))))
# dat_index <- dat_hclust$order
# 
# sub_dat <- sub_dat[, dat_index]
# sub_celltype <- cell_type[rand_ind]
sub_cluster_labels <- as.numeric(as.factor(sub_celltype))

dim(sub_dat)

## [1] 1010  500

Apply SAVER

# saver_dat <- SAVER::saver(t(as.matrix(sub_dat)))

saver_mat <- t(saver_dat$estimate)

2-1. Dependency Measures

library(reshape2) # melt function
library(ggplot2) # ggplot function
library(pcaPP) # Fast Kendall function
library(energy) # Distance Correlation
library(Hmisc) # Hoeffding's D measure

## Loading required package: lattice

## Loading required package: survival

## Loading required package: Formula

## 
## Attaching package: 'Hmisc'

## The following objects are masked from 'package:dplyr':
## 
##     src, summarize

## The following object is masked from 'package:SeuratObject':
## 
##     Key

## The following object is masked from 'package:Seurat':
## 
##     Key

## The following objects are masked from 'package:base':
## 
##     format.pval, units

library(zebu) # Normalized Mutual Information
# library(minerva) # Maximum Information Coefficient
library(XICOR) # Chatterjee's Coefficient
# library(dHSIC) # Hilbert Schmidt Independence Criterion
library(VineCopula) # Blomqvist's Beta

make_cormat <- function(dat_mat){
matrix_dat <- matrix(nrow = ncol(dat_mat), ncol = ncol(dat_mat))
  cor_mat_list <- list()
  
  basic_cor <- c("pearson", "spearman")
  # find each of the correlation matrices with Pearson, Spearman, Kendall Correlation Coefficients
  for (i in 1:2){
    print(i)
    cor_mat <- stats::cor(dat_mat, method = basic_cor[i])
    cor_mat[upper.tri(cor_mat, diag = T)] <- NA
    cor_mat_list[[i]] <- cor_mat
  }
  
  # functions that take matrix or data.frame as input
  no_loop_function <- c(pcaPP::cor.fk, Hmisc::hoeffd, 
                        VineCopula::BetaMatrix)
  for (i in 3:5){
    print(i)
    fun <- no_loop_function[[i-2]]
    cor_mat <- fun(dat_mat)
    if (i == 4){ # Hoeffding's D
      cor_mat <- cor_mat$D
    }
    cor_mat[upper.tri(cor_mat, diag = T)] <- NA
    cor_mat_list[[i]] <- cor_mat
  }
  
  # functions that take two variables as input to calculate correlations.
  need_loop <- c(zebu::lassie, energy::dcor2d, XICOR::calculateXI)

  for (i in 6:8){
    print(i)
    fun <- need_loop[[i-5]]
    
    cor_mat <- matrix(nrow = ncol(dat_mat),
                      ncol = ncol(dat_mat))
    
    for (j in 2:ncol(dat_mat)){
      for (k in 1:(j-1)){
        if (i == 6){
          cor_mat[j, k] <- fun(cbind(dat_mat[, j], dat_mat[, k]), continuous=c(1,2), breaks = 6, measure = "npmi")$global

        } else {
          cor_mat[j, k] <- fun(as.numeric(dat_mat[, j]),
                               as.numeric(dat_mat[, k]))
        }
      }
    }
    
    cor_mat[upper.tri(cor_mat, diag = T)] <- NA
    cor_mat_list[[i]] <- cor_mat
  }
  return(cor_mat_list)
}

draw_heatmap <- function(cor_mat){
    len <- 6
    melted_cormat <- melt(cor_mat)
    melted_cormat <- melted_cormat[!is.na(melted_cormat$value),]
    break_vec <- round(as.numeric(quantile(melted_cormat$value,
                                           probs = seq(0, 1, length.out = len),
                                           na.rm = T)),
                       4)
    break_vec[1] <- break_vec[1]-1
    break_vec[len] <- break_vec[len]+1
    melted_cormat$value <- cut(melted_cormat$value, breaks = break_vec)
    heatmap_color <- unique(melted_cormat$value)
  
    heatmap <- ggplot(data = melted_cormat, aes(x = Var2, y = Var1, fill = value))+
      geom_tile(colour = "Black") +
      ggplot2::scale_fill_manual(breaks = sort(heatmap_color), 
                                 values = rev(scales::viridis_pal(begin = 0, end = 1)
                                              (length(heatmap_color)))) +
      theme_bw() + # make the background white
      theme(panel.border = element_blank(), panel.grid.major = element_blank(),
            panel.grid.minor = element_blank(), axis.ticks = element_blank(),
            # erase tick marks and labels
            axis.text.x = element_blank(), axis.text.y = element_blank())
    
    return (heatmap)
}

make_cor_heatmap <- function(dat_mat, cor_mat_list){
  fun_lable <- c("Pearson's Correlation", "Spearman's Correlation", "Kendall's Correlation",
                 "Hoeffding's D", "Blomqvist's Beta", "NMI",
                  "Distance Correlation", "XI Correlation")
  
  cor_heatmap_list <- list()
  cor_abs_heatmap_list <- list()
  
  # make correlation matrices
  #cor_mat_list <- make_cormat(dat_mat)
  
  for (i in 1:8){
    print(i)
    if (i == 5){
      next
    }
    cor_mat <- abs(cor_mat_list[[i]])
    
    # get heatmaps
    cor_heatmap <- draw_heatmap(cor_mat)
    
    # ggplot labels
    ggplot_labs <- labs(title = paste("Heatmap of", fun_lable[i]),
                      x = "",
                      y = "",
                      fill = "Coefficient") # change the title and legend label
      
    cor_heatmap_list[[i]] <- cor_heatmap + ggplot_labs
    
    if (i %in% c(1,2,3,4,6)){
      cor_abs_mat <- abs(cor_mat_list[[i]])
      cor_abs_heatmap <- draw_heatmap(cor_abs_mat)
      ggplot_abs_labs <- labs(title = paste("Abs Heatmap of", fun_lable[i]),
                              x = "", # change the title and legend label
                              y = "", 
                              fill = "Coefficient") 
      cor_abs_heatmap_list[[i]] <- cor_abs_heatmap + ggplot_abs_labs
    } else {
      ggplot_abs_labs <- labs(title = paste("Abs Heatmap of", fun_lable[i]),
                              subtitle = "Equivalent to Non-Abs Heatmap",
                              x = "", # change the title and legend label
                              y = "", 
                              fill = "Coefficient") 
      cor_abs_heatmap_list[[i]] <- cor_heatmap + ggplot_abs_labs
    }
  }
  
  ans <- list(cor_heatmap_list, cor_abs_heatmap_list)
  
  return (ans)
}

load("SAVER_seurat_corr.RData")

# save(cormat_list, heatmap_list, saver_mat, sub_celltype, store_date, file = "SAVER_seurat_corr.RData")

# cormat_list <- make_cormat(saver_mat)
# heatmap_list <- make_cor_heatmap(saver_mat, cormat_list)

cor_pearson_mat <- cormat_list[[1]]; cor_spearman_mat <- cormat_list[[2]];
cor_kendall_mat <- cormat_list[[3]]; cor_hoeffd_mat <- cormat_list[[4]];
cor_blomqvist_mat <- cormat_list[[5]]; cor_MI_mat <- cormat_list[[6]];
cor_dist_mat <- cormat_list[[7]]; cor_XI_mat <- cormat_list[[8]];

1. Pearson’s correlation coefficient

• Pearson’s correlation is to measure linear dependency of data, X and Y
• \(-1 \leq \rho_{Pearson}(X, Y) \leq 1\)
• \(\rho_{Pearson}(X, Y) = \frac{\sum(x_i-\bar{x})(y_i -\bar{y})}{\sum(x_i-\bar{x})^2(y_i -\bar{y})^2}\)

cor_pearson_mat[1:5,1:5]

##              TMSB4X     MT-ND2    MT-ATP6     MT-CYB RPS25
## TMSB4X           NA         NA         NA         NA    NA
## MT-ND2  -0.02940452         NA         NA         NA    NA
## MT-ATP6  0.05764577  0.6005007         NA         NA    NA
## MT-CYB   0.01728402  0.6175195  0.6597476         NA    NA
## RPS25   -0.45423356 -0.1214727 -0.1986036 -0.1419443    NA

quantile(cor_pearson_mat, na.rm = T)

##          0%         25%         50%         75%        100% 
## -0.72766171 -0.08780746  0.02768556  0.14230112  0.96059754

quantile(abs(cor_pearson_mat), na.rm = T)

##           0%          25%          50%          75%         100% 
## 3.489797e-07 5.478188e-02 1.171753e-01 2.024889e-01 9.605975e-01

# plot the smallest correlations
cor_pearson_vec <- sort(abs(cor_pearson_mat), decreasing = T)
plot(cor_pearson_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_pearson_mat) == cor_pearson_vec[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_pearson_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_pearson_mat) == rev(cor_pearson_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_pearson_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[1]]

2. Spearman’s correlation coefficient

• It captures the monotonic relationship between data, X and Y
• \(-1 \leq \rho_{Spearman}(X,Y) \leq 1\)
• \(\rho_{Spearman} = 1 - \frac{6\sum{d_i^2}}{n(n^2-1)}\) where \(d_i\) is the difference between the ranks of \(x_i\) and \(y_i\)

cor_spearman_mat[1:5,1:5]

##              TMSB4X     MT-ND2    MT-ATP6    MT-CYB RPS25
## TMSB4X           NA         NA         NA        NA    NA
## MT-ND2   0.03179945         NA         NA        NA    NA
## MT-ATP6 -0.05865502  0.5912240         NA        NA    NA
## MT-CYB  -0.03411814  0.5749782  0.6511620        NA    NA
## RPS25   -0.11125302 -0.1000161 -0.1893696 -0.109827    NA

quantile(cor_spearman_mat, na.rm = T)

##          0%         25%         50%         75%        100% 
## -0.84618253 -0.07354277  0.02984100  0.13308323  0.81889279

quantile(abs(cor_spearman_mat), na.rm = T)

##           0%          25%          50%          75%         100% 
## 1.613408e-06 4.973064e-02 1.053442e-01 1.810083e-01 8.461825e-01

# plot the smallest correlations
cor_spearman_vec <- sort(abs(cor_spearman_mat), decreasing = T)
plot(cor_spearman_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_spearman_mat) == cor_spearman_vec[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_spearman_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_spearman_mat) == rev(cor_spearman_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_spearman_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[2]]

3. Kendall’s Tau

• It is an alternative method to Spearman’s correlations, identifying monotonic relationships.
• \(-1 \leq \rho_{Kendall}(X,Y) \leq 1\)
• \(\rho_{Kendall}(X,Y) = \frac{\#\;concordant\;pairs - \#\;discordant \;pairs}{0.5n(n-1)}\)

cor_kendall_mat[1:5,1:5]

##              TMSB4X      MT-ND2    MT-ATP6      MT-CYB RPS25
## TMSB4X           NA          NA         NA          NA    NA
## MT-ND2   0.01799150          NA         NA          NA    NA
## MT-ATP6 -0.04475093  0.43496159         NA          NA    NA
## MT-CYB  -0.02276791  0.40848512  0.4764382          NA    NA
## RPS25   -0.07237012 -0.06244924 -0.1186702 -0.07486883    NA

quantile(cor_kendall_mat, na.rm = T)

##          0%         25%         50%         75%        100% 
## -0.66004350 -0.05001927  0.02019916  0.08993943  0.63332765

quantile(abs(cor_kendall_mat), na.rm = T)

##           0%          25%          50%          75%         100% 
## 3.962028e-06 3.370566e-02 7.132955e-02 1.228110e-01 6.600435e-01

# plot the smallest correlations
cor_kendall_vec <- sort(abs(cor_kendall_mat), decreasing = T)
plot(cor_kendall_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_kendall_mat) == cor_kendall_vec[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_kendall_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_kendall_mat) == rev(cor_kendall_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_kendall_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[3]]

4. Hoeffding’s D

• It tests the independence of data by calculating the distance between the product of the marginal distributions under the null hypothesis and the empirical bi-variate distribution.
• \(-1 \leq D(X,Y) \leq 1\)
• \(D(X,Y) = \frac{(n-2)(n-3)D_1+D_2-2(n-2)D_3}{n(n-1)(n-2)(n-3)(n-4)}\)
  • \(D_1 = \sum_{i=1}^{n} Q_i(Q_i-1)\)
  • \(D_2 = \sum_{i=1}^{n} (R_i-1)(R_i-2)(S_j-1)(S_j-2)\)
  • \(D_3 = \sum_{i=1}^{n} (R_i-2)(S_i-2)Q_i\)

cor_hoeffd_mat[1:5,1:5]

##              TMSB4X      MT-ND2    MT-ATP6      MT-CYB RPS25
## TMSB4X           NA          NA         NA          NA    NA
## MT-ND2  0.009290504          NA         NA          NA    NA
## MT-ATP6 0.011479529 0.135400141         NA          NA    NA
## MT-CYB  0.006091448 0.109296103 0.15678142          NA    NA
## RPS25   0.008666603 0.007664955 0.01265554 0.005907249    NA

quantile(cor_hoeffd_mat, na.rm = T)

##           0%          25%          50%          75%         100% 
## 0.0006317592 0.0052005399 0.0082638904 0.0145437026 0.3361010002

# plot the smallest correlations
cor_hoeffd_vec <- sort(abs(cor_hoeffd_mat), decreasing = T)
plot(cor_hoeffd_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_hoeffd_mat) == cor_hoeffd_vec[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_hoeffd_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_hoeffd_mat) == rev(cor_hoeffd_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_hoeffd_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[4]]

5. Blomqvist’s Beta

• It measures dependency between variables by constructing a two-way contingency table with the medians of each margin as cutting points.
• \(0 \leq \beta \leq 1\)
• \(\beta_n = \frac{n_1-n_2}{n_1+n_2} = \frac{2n_1}{n_1+n_2} - 1\)
• \(\beta = P\{(X-\tilde{x})(Y-\tilde{y})>0\} - P\{(X-\tilde{x})(Y-\tilde{y}) < 0\}\)

cor_blomqvist_mat[1:5,1:5]

##      [,1] [,2] [,3] [,4] [,5]
## [1,]   NA   NA   NA   NA   NA
## [2,]    1   NA   NA   NA   NA
## [3,]    1    1   NA   NA   NA
## [4,]    1    1    1   NA   NA
## [5,]    1    1    1    1   NA

quantile(cor_blomqvist_mat, na.rm = T)

##        0%       25%       50%       75%      100% 
## 0.9881188 1.0000000 1.0000000 1.0000000 1.0000000

quantile(abs(cor_blomqvist_mat), na.rm = T)

##        0%       25%       50%       75%      100% 
## 0.9881188 1.0000000 1.0000000 1.0000000 1.0000000

# plot the smallest correlations
cor_blomqvist_vec <- sort(abs(cor_blomqvist_mat), decreasing = T)
plot(cor_blomqvist_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(unique(abs(cor_blomqvist_mat)) == unique(cor_blomqvist_vec)[i], arr.ind = T)
 idx1 <- idx[1,1]; idx2 <- idx[1,2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_blomqvist_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_blomqvist_mat) == rev(cor_blomqvist_vec)[i], arr.ind = T)
 idx1 <- idx[1,1]; idx2 <- idx[1,2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_blomqvist_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[5]]

## NULL

6. Normalized Mutual Information

• It measures how much one random variable gives information about the other. For example, High mutual information indicates a large reduction in uncertainty.
• \(0 \leq MI(X,Y) \leq 1\), as it is normalized.
• $MI(X,Y) = f_{X,Y} (x,y) log_2 ; dxdy $
• \(MI(X,Y) = \sum \sum p_{X,Y} (x,y) log \frac{p_{X,Y} (x,y)}{P_X(x)P_Y(y)}\)

cor_MI_mat[1:5,1:5]

##           [,1]      [,2]      [,3]      [,4] [,5]
## [1,]        NA        NA        NA        NA   NA
## [2,] 0.5900765        NA        NA        NA   NA
## [3,] 0.6587018 0.2397348        NA        NA   NA
## [4,] 0.7150763 0.2277529 0.3148607        NA   NA
## [5,] 0.9971672 0.1872224 0.0834614 0.1378969   NA

quantile(cor_MI_mat, na.rm = T)

##         0%        25%        50%        75%       100% 
## 0.02008612 0.11215506 0.13350667 0.15808349 1.00000000

# plot the smallest correlations
cor_MI_vec <- sort(abs(cor_MI_mat), decreasing = T)
plot(cor_MI_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(unique(abs(cor_MI_mat)) == unique(cor_MI_vec)[i], arr.ind = T)
 idx1 <- idx[1,1]; idx2 <- idx[1,2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_MI_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_MI_mat) == rev(cor_MI_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_MI_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[6]]

7. Distance Correlation

• it is measure to identify non-linear relationships between two random variables with energy distances.
• distance correlation is calculated by dividing the distance covariance between X and Y by the product of their distance standard deviations.
• \(0 \leq dCor \leq 1\)
• \(dCor(X,Y) = \frac{dCov(Y,Y)}{\sqrt{dVar(X)dVar(Y)}}\)
  • \(dCov(X, Y) = \sqrt{\frac{1}{n^2} \sum_{k=1, l=1}^{n} A_{k,l}B_{k,l}}\)
  • \(dVar(X) = dCov(X,X) and dVar(Y) = dCov(Y, Y)\)

cor_dist_mat[1:5,1:5]

##            [,1]       [,2]       [,3]       [,4] [,5]
## [1,]         NA         NA         NA         NA   NA
## [2,] 0.03183111         NA         NA         NA   NA
## [3,] 0.04054841 0.30943786         NA         NA   NA
## [4,] 0.05429602 0.31399171 0.38388541         NA   NA
## [5,] 0.19099907 0.03445755 0.05239319 0.03176544   NA

quantile(cor_dist_mat, na.rm = T)

##         0%        25%        50%        75%       100% 
## 0.00710533 0.02739638 0.03960727 0.06265141 0.81891030

# plot the smallest correlations
cor_dist_vec <- sort(abs(cor_dist_mat), decreasing = T)
plot(cor_dist_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_dist_mat) == cor_dist_vec[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_dist_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_dist_mat) == rev(cor_dist_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_dist_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[7]]

8. Chatterjee’s XI Correlation

• It measures the degree of dependence between the variables with concept of rank.
• \(0 \leq \xi_n \leq 1\)
• \(\xi_n(X,Y) = 1 - \frac{3\sum_{i=1}^{n-1} |r_{i+1} -r_i|}{n^2-1}\)
• \(\xi_n(X,Y) = 1 - \frac{n\sum_{i=1}^{n-1}|r_{i+1}-r_i}{2\sum_{i=1}^{n}l_i(n-l_i)}\)

cor_XI_mat[1:5,1:5]

##           [,1]      [,2]      [,3]     [,4] [,5]
## [1,]        NA        NA        NA       NA   NA
## [2,] 0.7929436        NA        NA       NA   NA
## [3,] 0.7875062 0.8490886        NA       NA   NA
## [4,] 0.7943859 0.8358267 0.8524429       NA   NA
## [5,] 0.7977288 0.7724003 0.8006900 0.787246   NA

quantile(cor_XI_mat, na.rm = T)

##        0%       25%       50%       75%      100% 
## 0.7090786 0.7767731 0.7857653 0.7944726 0.8933626

# plot the smallest correlations
cor_XI_vec <- sort(abs(cor_XI_mat), decreasing = T)
plot(cor_XI_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_XI_mat) == cor_XI_vec[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_XI_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
 idx <- which(abs(cor_XI_mat) == rev(cor_XI_vec)[i], arr.ind = T)
 idx1 <- idx[1]; idx2 <- idx[2]
 
 plot(saver_mat[,idx1], saver_mat[,idx2], col = sub_cluster_labels, asp = T,
      pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
      ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"), 
      main = paste0("Correlation of ", round(cor_XI_mat[idx1, idx2], 3)))
}

Heatmap

heatmap_list[[1]][[8]]

quantile_mat <- c()
for (i in 1:length(cormat_list)){
  quantile_mat <- rbind(quantile_mat,
                        quantile(abs(cormat_list[[i]]), probs = c(0.10, 0.95), na.rm = T))
}
rownames(quantile_mat) <- c("Pearson", "Spearman", "Kendall", "Hoeffding's D",
                            "Blomqvist's Beta","Dist. Corr", "NMI", "XI Corr")

quantile_mat

##                          10%        95%
## Pearson          0.021712468 0.35913737
## Spearman         0.019754490 0.31182419
## Kendall          0.013397520 0.21416748
## Hoeffding's D    0.003628521 0.03430303
## Blomqvist's Beta 1.000000000 1.00000000
## Dist. Corr       0.094633295 0.20696296
## NMI              0.020866540 0.12856267
## XI Corr          0.768451107 0.80685938

# save(quantile_mat, store_date, file = "SAVER_seurat_quantile.RData")